Problem 65 - Algebra for College Students

Question

Solve each problem by setting up and solving an appropriate inequality. A person's intelligence quotient \((I)\) is found by dividing mental age \((M)\), as indicated by standard tests, by chronological age \((C)\) and then multiplying this ratio by 100 . The formula \(I=\frac{100 M}{C}\) can be used. If the \(I\) range of a group of 11-year-olds is given by \(80 \leq I \leq 140\), find the range of the mental age of this group.

Step-by-Step Solution

Answer

The range of mental age is 8.8 to 15.4 years.

Step 1: Understand the Problem

We need to determine the range of mental age \(M\) for a group of 11-year-olds, given the intelligence quotient \(I\) range of \(80 \leq I \leq 140\) and the formula \(I = \frac{100M}{C}\), where \(C = 11\).

Step 2: Formulate the Inequality for I

Since \(C = 11\), substitute \(C\) in the formula: \(I = \frac{100M}{11}\). We know that \(80 \leq I \leq 140\), which gives \(80 \leq \frac{100M}{11} \leq 140\).

Step 3: Solve for M in the Lower Bound

First, solve the inequality \(80 \leq \frac{100M}{11}\). Multiply both sides by 11 to get \(880 \leq 100M\). Next, divide both sides by 100 to find \(M\): \(8.8 \leq M\).

Step 4: Solve for M in the Upper Bound

Now, solve the inequality \(\frac{100M}{11} \leq 140\). Multiply both sides by 11 to get \(100M \leq 1540\). Then, divide both sides by 100 to solve for \(M\): \(M \leq 15.4\).

Step 5: Determine the Range of M

Combine the results from Steps 3 and 4 to get the range for \(M\). Thus, the mental age \(M\) satisfies \(8.8 \leq M \leq 15.4\).

Key Concepts

Understanding Intelligence Quotient (IQ)

Intelligence Quotient, or IQ, is a measure used to assess human intelligence. You might have heard about IQ being a score derived from standardized tests designed to evaluate human intellectual ability. But how exactly is it measured?
IQ is determined by the formula:

• \(I = \frac{100M}{C}\)

in which \(M\) represents the mental age and \(C\) the chronological age. This means that an individual's IQ is calculated by dividing their mental age by their chronological age and then multiplying the result by 100. This formula helps standardize IQ scores so that they can be compared across different ages and populations.
Understanding IQ requires knowing two essential concepts: mental age and chronological age.
By measuring IQ, we gain insights into a person's cognitive abilities in comparison to others in their age group.

Decoding Mental Age

Mental age, a critical component in calculating IQ, refers to the age level at which an individual is functioning intellectually. It is derived from standardized intelligence tests that assess different cognitive abilities such as reasoning, memory, and problem-solving.
When someone talks about a person's mental age, they mean how well they perform intellectually compared to typical expectations of their chronological peers. For example, if an 11-year-old child performs on cognitive tests similar to an average 13-year-old, their mental age is considered to be 13.
Mental age plays a crucial role because it directly influences the IQ formula:

• \(I = \frac{100M}{C}\)

where an increase in mental age generally results in a higher IQ, assuming the chronological age remains constant.

Defining Chronological Age

Chronological age is much simpler to comprehend compared to mental age. It is the actual age of an individual, calculated from their date of birth to the present time. In the context of IQ, chronological age is used as a reference point for the mental age.
Understanding chronological age is straightforward. If you are dealing with a group of 11-year-olds, as in the example problem, then the chronological age, \(C\), is simply 11. The standardization in the IQ formula comes from using this chronological age as the denominator:

• \(I = \frac{100M}{C}\)

Therefore, any variation in IQ results mainly from variations in mental age, given that chronological age is consistent and known.

Mastering Formula Manipulation

Mastering formula manipulation is key to solving problems involving IQ. Let's break down how to manipulate the formula:

• \(I = \frac{100M}{C}\)

into inequalities for defining ranges.
Given the formula, if you know certain values like the IQ range and chronological age, you can solve for the mental age. For example, in the original exercise, the goal was to find the range of mental ages \(M\) for a given IQ range: \(80 \leq I \leq 140\). By substituting the known values and manipulating the inequalities, you derive \ the mental age range:

• \(8.8 \leq M \leq 15.4\)

Formula manipulation involves basic algebraic operations like multiplication, division, and understanding inequalities to isolate the desired variable. Such manipulation is crucial in accurately setting up the problem and finding solutions.